`BookProof.ChapterAttentionCapacity.scoreSoftmax_distScore_ge_of_separated` {k : Fin m → EuclideanSpace ℝ (Fin n)} {r beta : ℝ} (hb : 0 ≤ beta) (hr : 0 ≤ r) (hsep : keysSeparated k

Proved

BookProof.ChapterAttentionCapacity.scoreSoftmax_distScore_ge_of_separated

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionCapacity.

BookProof.ChapterAttentionCapacity.scoreSoftmax_distScore_ge_of_separated {k : Fin m → EuclideanSpace ℝ (Fin n)} {r beta : ℝ} (hb : 0 ≤ beta) (hr : 0 ≤ r) (hsep : keysSeparated k r) (i : Fin m) : 1 / (1 + ((m : ℝ) - 1) * Real.exp (-(beta * r ^ 2))) ≤ scoreSoftmax beta (distScore (k i) k) i

Formalization note: Lean 4 identifier BookProof.ChapterAttentionCapacity.scoreSoftmax_distScore_ge_of_separated.

Preamble

-- Generated from ChapterAttentionCapacity.lean — theorem BookProof.ChapterAttentionCapacity.scoreSoftmax_distScore_ge_of_separated
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionCapacity
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionCapacity


open scoped BigOperators

open Filter Topology

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m n : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

Formal statement

theorem BookProof.ChapterAttentionCapacity.scoreSoftmax_distScore_ge_of_separated {k : Fin m → EuclideanSpace ℝ (Fin n)}
    {r beta : ℝ} (hb : 0 ≤ beta) (hr : 0 ≤ r) (hsep : keysSeparated k r) (i : Fin m) :
    1 / (1 + ((m : ℝ) - 1) * Real.exp (-(beta * r ^ 2)))
      ≤ scoreSoftmax beta (distScore (k i) k) i := by sorry

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